Horizontal Centripetal Force

Imagine you are spinning a weight at the end of a string horizontally. As one increases the speed with which the weight is being spinned, the motion becomes more and more horizontal. I understand this occurs because the centripetal force (tension of the string) is increasing but at the same time the mg of the weight remains constant, thus the "sag" of the string disappears. My question ,though, is, Why does the motion never become truly horizontal?

The motion is horizontal, the weight will stay at the same height, but the string can't be horizontal because the force of the string would be horizontal, and you'd still have the force of gravity downwards, so you get a net force downwards, so the acceleration of the weight would have a downwards component as well, so it would move downwards, and then the string would no longer be horizontal